India is culturally diverse, more diverse than Europe by some measures. Just as it is difficult to make statements about `European' practices which hold true for Basques, Czechs, English, Finnish, French, Germans, Greeks, Hungarians, and Norwegians, just to take a small subset, it is equally difficult to make statements about Indians.
The use of last name itself, is not very old in some regions, especially for women. Today, most people in North India do go by a first name followed by a last name, though sometimes the first name consists of a conjoined set of words (e.g. mani gopal), and sometimes the same holds for the last name (e.g. sen gupta). There are no fixed conventions whether these are written as separate words or joined when writing in the Roman script (e.g. manigopal), and some people will write it as one word, but still use two initials when the full names are not desired!
The last name can be followed by or replaced by an honorific or occupational title, and sometimes becomes a last name. Since the last name gives away the `caste' to the cognoscenti, some people drop it altogether, and go only by the first names. The honorific if placed before the first name is usually recognized as such. In some regions, a word signifying status like `unmarried woman', or `belonging to the Sikh religion', is interposed between the first name and the last name, and sometimes the last name is entirely omitted in this context. In some other regions, a woman is often addressed by her first name followed by a word meaning `goddess'.
There are large swaths of South India where there is no last name but the first name is preceded by the father's name, and, I know of at least one woman who replaced that by her mother's name instead. The father's name in this context is usually denoted by just the initial. I know of one person who insisted on writing this initial after his first name, so I guess it couldn't have been called an initial.
Other markers like village or tribe name are often prepended, appended or infixed, especially in the Western regions. I knew a guy once who called himself: Madhabhushi Venkata Narasimha Sesha Pundarikaksha Madhava Ravikumar, and though I know the derivation of most of those (I may not have gotten the spellings right, since the conventional spellings in the Roman script varies across India), I never though to ask why he had so many of them: it is just too common in some parts of India to have multiple names. In this case, he went by M.V.N.S.P.M. Ravikumar and most people called him Ravikumar (and maybe other names, since in most of India the pet names are often completely unrelated to the formal names); actually many people called him AtoZed even though not all the letters actually appear in his name!